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  前向星与链式前向星
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  <h1 id="前向星"><a href="#前向星" class="headerlink" title="前向星"></a>前向星</h1><p>前向星是以存储边的方式来存储图，先将边读入并存储在连续的数组中，然后按照边的起点进行排序，这样数组中起点相同的边就能够在数组中进行连续访问了。<br>其有点在于实现简单，易于理解。缺点是在读入边后需要对所有边进行一次排序，带来了时间开销，实用性也较差。因此只适合离线算法。  </p>
<h1 id="链式前向星（实质为数组模拟链表）"><a href="#链式前向星（实质为数组模拟链表）" class="headerlink" title="链式前向星（实质为数组模拟链表）"></a>链式前向星（实质为数组模拟链表）</h1><p>链式前向星和邻接表类似，也是链式结构的结合，每个节点i均有一个链表，此链表内的数据均为以节点i为起点的所有边的集合（对比邻接表存的是顶点的集合），边的表示为一个四元组(to,w,next)，其中to代表该条边的有向点对(w,v)，w代表边上的权值，next指向下一条边。<br>调用的时候只要通过head[i]就能访问到由节点i出发的第一条边的编号，通过编号到edge数组进行索引可以得到边的具体信息，然后根据这条边的next域可以得到第二条边的编号，以此类推，直到next域为INF（这里的INF即head数组初始化的值，一般取-1即可）</p>
<span id="more"></span>
<h1 id="链式前向星基本原理"><a href="#链式前向星基本原理" class="headerlink" title="链式前向星基本原理"></a>链式前向星基本原理</h1><p>链式前向星是以边为主的存图方式，因此我们需要利用结构体来规划边，这会使图变得更加清晰。  </p>
<h2 id="结构"><a href="#结构" class="headerlink" title="结构"></a>结构</h2><p>这里注意下面两个必要的数组： </p>
<pre><code>Edge edge[edgenumber];    //结构体数组edge存边，edge[i]表示编号为i的边  
int head[vexnumber];    //head[i]存以节点i为起点的第一条边（在edge中的编号）
</code></pre><p>这里实现的是<br><figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">struct</span> <span class="title class_">Edge</span>&#123;</span><br><span class="line">	<span class="type">int</span> next;   <span class="comment">//下一条边的存储下标(默认0) </span></span><br><span class="line">	<span class="type">int</span> to;     <span class="comment">//某个节点u的邻接点</span></span><br><span class="line">	<span class="type">int</span> w;      <span class="comment">//权值 </span></span><br><span class="line">&#125;; </span><br><span class="line">Edge edge[<span class="number">500010</span>];</span><br></pre></td></tr></table></figure></p>
<h3 id="Edge结构体不用记录顶点u"><a href="#Edge结构体不用记录顶点u" class="headerlink" title="Edge结构体不用记录顶点u"></a>Edge结构体不用记录顶点u</h3><p>如上所述，前向星存储的是每个顶点的邻接边。可以知道，在每个顶点后面连接一个链表，不过链表的节点表示的是邻接边而非邻接点。而用数组实现这个链表，就是前向星。<br>无论采用哪种结构存图，都应该是每读入一条边的信息，就更新一次图的结构。读入一条边u—&gt;v，那么我们在存储图的时候，对于顶点u,该边就是u的一条邻接边，我们就把这条边“挂”在u后面即可。等读入所有的边，建图完成。<br>顶点u后“挂”的边就全是顶点u的邻接边，因此只需要记录这些邻接边的另一个顶点，而不需要存储顶点u。<br>由于边是按顺序一条一条读入，我们就很自然的想到对边进行编号，然后将这些边读入edge数组。对于Edge结构体中的next，其实就相当于链表中的next指针。不过此处的next是int型变量，存储的是下一条邻接边的编号。  </p>
<h3 id="head数组的用处"><a href="#head数组的用处" class="headerlink" title="head数组的用处"></a>head数组的用处</h3><p>head数组相当于邻接链表中的表头数据，head[u]就表示顶点u的某一条邻接边（其编号）。根据这条边的next，就能找出顶点u所有的邻接边。  </p>
<h2 id="增边"><a href="#增边" class="headerlink" title="增边"></a>增边</h2><p>若以点i为起点的边新增了一条，在edge中的下标为j。那么edge[j].next = head[i]，此后head[i] = j<br>即每次新加的边作为第一条边，最后倒序遍历  </p>
<figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="type">void</span> <span class="title">Add</span><span class="params">(<span class="type">int</span> u, <span class="type">int</span> v, <span class="type">int</span> w)</span> </span>&#123;  <span class="comment">//起点u, 终点v, 权值w </span></span><br><span class="line">	<span class="comment">//cnt为边的计数，从1开始计 </span></span><br><span class="line">	edge[++cnt].next = head[u];</span><br><span class="line">	edge[cnt].w = w;</span><br><span class="line">	edge[cnt].to = v;</span><br><span class="line">	head[u] = cnt;    <span class="comment">//第一条边为当前边 </span></span><br><span class="line">&#125; </span><br></pre></td></tr></table></figure>
<h2 id="遍历"><a href="#遍历" class="headerlink" title="遍历"></a>遍历</h2><p>遍历以st为起点的边<br>    for(int i = head[st]; i != -1; i = edge[i].next)<br>i为开始的第一条边，每次指向下一条，以初始化的-1为结束标志  </p>
<h1 id="简单案例"><a href="#简单案例" class="headerlink" title="简单案例"></a>简单案例</h1><figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">#<span class="keyword">include</span> <span class="string">&lt;iostream&gt;</span></span></span><br><span class="line"><span class="keyword">using</span> <span class="keyword">namespace</span> std;</span><br><span class="line"> </span><br><span class="line"><span class="meta">#<span class="keyword">define</span> MAXM 500000 + 10</span></span><br><span class="line"><span class="meta">#<span class="keyword">define</span> MAXN 10000 + 10</span></span><br><span class="line"> </span><br><span class="line"><span class="keyword">struct</span> <span class="title class_">Edge</span>&#123;</span><br><span class="line">	<span class="type">int</span> next;   <span class="comment">//下一条边的存储下标 </span></span><br><span class="line">	<span class="type">int</span> to;     <span class="comment">//这条边的终点 </span></span><br><span class="line">	<span class="type">int</span> w;      <span class="comment">//权值 </span></span><br><span class="line">&#125;; </span><br><span class="line">Edge edge[MAXM];</span><br><span class="line"> </span><br><span class="line"><span class="type">int</span> n, m, cnt;</span><br><span class="line"><span class="type">int</span> head[MAXN];  <span class="comment">//head[i]表示以i为起点的第一条边 </span></span><br><span class="line"> </span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">add</span><span class="params">(<span class="type">int</span> u, <span class="type">int</span> v, <span class="type">int</span> w)</span> </span>&#123;  <span class="comment">//起点u, 终点v, 权值w </span></span><br><span class="line">	edge[++cnt].next = head[u];</span><br><span class="line">	edge[cnt].w = w;</span><br><span class="line">	edge[cnt].to = v;</span><br><span class="line">	head[u] = cnt;    <span class="comment">//第一条边为当前边 </span></span><br><span class="line">&#125; </span><br><span class="line"> </span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">print</span><span class="params">()</span> </span>&#123;</span><br><span class="line">	<span class="type">int</span> st;</span><br><span class="line">	cout &lt;&lt; <span class="string">&quot;Begin with[Please Input]: \n&quot;</span>;</span><br><span class="line">	cin &gt;&gt; st;</span><br><span class="line">	<span class="keyword">for</span>(<span class="type">int</span> i = head[st]; i != <span class="number">0</span>; i = edge[i].next) &#123;</span><br><span class="line">		<span class="comment">//i开始为第一条边，每次指向下一条(以0为结束标志)若下标从0开始，next应初始化-1 </span></span><br><span class="line">		cout &lt;&lt; <span class="string">&quot;Start: &quot;</span> &lt;&lt; st &lt;&lt; endl;</span><br><span class="line">		cout &lt;&lt; <span class="string">&quot;End: &quot;</span> &lt;&lt; edge[i].to &lt;&lt; endl;</span><br><span class="line">		cout &lt;&lt; <span class="string">&quot;W: &quot;</span> &lt;&lt; edge[i].w &lt;&lt; endl &lt;&lt; endl; </span><br><span class="line">	&#125;</span><br><span class="line">&#125;</span><br><span class="line"> </span><br><span class="line"><span class="function"><span class="type">int</span> <span class="title">main</span><span class="params">()</span> </span>&#123;</span><br><span class="line">	<span class="type">int</span> s, t, w;</span><br><span class="line">	cin &gt;&gt; n &gt;&gt; m;</span><br><span class="line">	<span class="keyword">for</span>(<span class="type">int</span> i = <span class="number">1</span>; i &lt;= m; i++) &#123;</span><br><span class="line">		cin &gt;&gt; s &gt;&gt; t &gt;&gt; w;</span><br><span class="line">		<span class="built_in">add</span>(s, t, w);</span><br><span class="line">	&#125;</span><br><span class="line">	<span class="built_in">print</span>(); </span><br><span class="line">	<span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>
<h1 id="链式前向星实现SPFA"><a href="#链式前向星实现SPFA" class="headerlink" title="链式前向星实现SPFA"></a>链式前向星实现SPFA</h1><figure class="highlight c++"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br><span class="line">77</span><br><span class="line">78</span><br><span class="line">79</span><br><span class="line">80</span><br><span class="line">81</span><br><span class="line">82</span><br><span class="line">83</span><br><span class="line">84</span><br><span class="line">85</span><br><span class="line">86</span><br><span class="line">87</span><br><span class="line">88</span><br><span class="line">89</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">#<span class="keyword">include</span><span class="string">&lt;iostream&gt;</span></span></span><br><span class="line"><span class="meta">#<span class="keyword">include</span><span class="string">&lt;cstdio&gt;</span></span></span><br><span class="line"><span class="meta">#<span class="keyword">include</span><span class="string">&lt;queue&gt;</span></span></span><br><span class="line"><span class="keyword">using</span> <span class="keyword">namespace</span> std;</span><br><span class="line"></span><br><span class="line"><span class="meta">#<span class="keyword">define</span> MAXM 50000 + 10</span></span><br><span class="line"><span class="meta">#<span class="keyword">define</span> MAXN 10000 + 10</span></span><br><span class="line"><span class="meta">#<span class="keyword">define</span> INF 0x3f3f3f3f3f</span></span><br><span class="line"></span><br><span class="line"><span class="keyword">struct</span> <span class="title class_">Edge</span> &#123;</span><br><span class="line">	<span class="type">int</span> next;</span><br><span class="line">	<span class="type">int</span> to;</span><br><span class="line">	<span class="type">int</span> w;</span><br><span class="line">&#125;;</span><br><span class="line">Edge edge[MAXM];</span><br><span class="line"></span><br><span class="line"><span class="type">int</span> n, m, st, cnt;</span><br><span class="line"><span class="type">int</span> head[MAXN];</span><br><span class="line"></span><br><span class="line"><span class="type">int</span> d[MAXN];</span><br><span class="line"><span class="type">bool</span> inq[MAXN];</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">inline</span> <span class="type">int</span> <span class="title">Read</span><span class="params">()</span> </span>&#123;</span><br><span class="line">	<span class="type">char</span> c;</span><br><span class="line">	<span class="type">int</span> ans = <span class="number">0</span>;</span><br><span class="line">	<span class="type">bool</span> sign = <span class="literal">false</span>;</span><br><span class="line">	<span class="keyword">while</span> (!<span class="built_in">isdigit</span>(c = <span class="built_in">getchar</span>()) &amp;&amp; c != <span class="string">&#x27;-&#x27;</span>);</span><br><span class="line">	<span class="keyword">if</span> (c == <span class="string">&#x27;-&#x27;</span>) &#123;</span><br><span class="line">		sign = <span class="literal">true</span>;</span><br><span class="line">		c = <span class="built_in">getchar</span>();</span><br><span class="line">	&#125;</span><br><span class="line">	<span class="keyword">do</span> &#123;</span><br><span class="line">		ans = (ans &lt;&lt; <span class="number">3</span>) + (ans &lt;&lt; <span class="number">1</span>) + (c ^ <span class="string">&#x27;0&#x27;</span>);</span><br><span class="line">	&#125; <span class="keyword">while</span> (<span class="built_in">isdigit</span>(c = <span class="built_in">getchar</span>()));</span><br><span class="line">	<span class="keyword">return</span> sign ? -ans : ans;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">add</span><span class="params">(<span class="type">int</span> u, <span class="type">int</span> v, <span class="type">int</span> w)</span> </span>&#123;</span><br><span class="line">	edge[++cnt].next = head[u];</span><br><span class="line">	edge[cnt].to = v;</span><br><span class="line">	edge[cnt].w = w;</span><br><span class="line">	head[u] = cnt;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">read</span><span class="params">()</span> </span>&#123;</span><br><span class="line">	<span class="type">int</span> x, y, w;</span><br><span class="line">	n = <span class="built_in">Read</span>();</span><br><span class="line">	m = <span class="built_in">Read</span>();</span><br><span class="line">	st = <span class="built_in">Read</span>();</span><br><span class="line">	<span class="keyword">for</span> (<span class="type">int</span> i = <span class="number">1</span>; i &lt; m; i++) &#123;</span><br><span class="line">		x = <span class="built_in">Read</span>();</span><br><span class="line">		y = <span class="built_in">Read</span>();</span><br><span class="line">		w = <span class="built_in">Read</span>();</span><br><span class="line">		<span class="built_in">add</span>(x, y, w);</span><br><span class="line">	&#125;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="type">void</span> <span class="title">SPFA</span><span class="params">(<span class="type">int</span> x)</span> </span>&#123;</span><br><span class="line">	d[x] = <span class="number">0</span>;</span><br><span class="line">	<span class="keyword">for</span> (<span class="type">int</span> i = <span class="number">1</span>; i &lt; n; i++)</span><br><span class="line">		d[i] = INF;</span><br><span class="line">	queue&lt;<span class="type">int</span>&gt; q;</span><br><span class="line">	q.<span class="built_in">push</span>(x);</span><br><span class="line">	inq[x] = <span class="literal">true</span>;</span><br><span class="line">	<span class="keyword">while</span> (!q.<span class="built_in">empty</span>()) &#123;</span><br><span class="line">		<span class="type">int</span> k = q.front;</span><br><span class="line">		q.<span class="built_in">pop</span>();</span><br><span class="line">		inq[k] = <span class="literal">false</span>;</span><br><span class="line">		<span class="keyword">for</span> (<span class="type">int</span> i = head[k]; i != <span class="number">0</span>; i = edge[i].next) &#123;</span><br><span class="line">			<span class="type">int</span> j = edge[i].to;</span><br><span class="line">			<span class="keyword">if</span> (d[j] &gt; d[k] + edge[i].w) &#123;</span><br><span class="line">				d[j] = d[k] + edge[i].w;</span><br><span class="line">				<span class="keyword">if</span> (!inq[j]) &#123;</span><br><span class="line">					q.<span class="built_in">push</span>(j);</span><br><span class="line">					inq[j] = <span class="literal">true</span></span><br><span class="line">				&#125;</span><br><span class="line">			&#125;</span><br><span class="line">		&#125;</span><br><span class="line">	&#125;</span><br><span class="line">	<span class="keyword">for</span> (<span class="type">int</span> i = <span class="number">1</span>; i &lt;= n; i++)</span><br><span class="line">		<span class="built_in">printf</span>(<span class="string">&quot;%d &quot;</span>, d[i]);</span><br><span class="line">	<span class="built_in">printf</span>(<span class="string">&quot;\n&quot;</span>);</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="type">int</span> <span class="title">main</span><span class="params">()</span> </span>&#123;</span><br><span class="line">	<span class="built_in">read</span>();</span><br><span class="line">	<span class="built_in">SPFA</span>(st);</span><br><span class="line">	<span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>
<p>关于SPFA算法的详解可见<a href="/2020/02/13/intro-SPFA/" title="SPFA算法详解">SPFA算法详解</a> </p>
 
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        let $imgArr = document.querySelectorAll(('.article-entry img:not(.reward-img)'))

        $imgArr.forEach(($em, i) => {
            $em.onclick = () => {
                // slider展开状态
                // todo: 这样不好，后面改成状态
                if (document.querySelector('.left-col.show')) return
                let items = []
                $imgArr.forEach(($em2, i2) => {
                    let img = $em2.getAttribute('data-idx', i2)
                    let src = $em2.getAttribute('data-target') || $em2.getAttribute('src')
                    let title = $em2.getAttribute('alt')
                    // 获得原图尺寸
                    const image = new Image()
                    image.src = src
                    items.push({
                        src: src,
                        w: image.width || $em2.width,
                        h: image.height || $em2.height,
                        title: title
                    })
                })
                var gallery = new PhotoSwipe(pswpElement, PhotoSwipeUI_Default, items, {
                    index: parseInt(i)
                });
                gallery.init()
            }
        })
    }
    viewer_init()
</script>

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<link rel="stylesheet" href="/css/clipboard.css">

<script src="https://cdn.jsdelivr.net/npm/clipboard@2/dist/clipboard.min.js"></script>
<script>
  function wait(callback, seconds) {
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  }
  !function (e, t, a) {
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      var copyHtml = '';
      copyHtml += '<button class="btn-copy" data-clipboard-snippet="">';
      copyHtml += '<i class="ri-file-copy-2-line"></i><span>COPY</span>';
      copyHtml += '</button>';
      $(".highlight .code pre").before(copyHtml);
      $(".article pre code").before(copyHtml);
      var clipboard = new ClipboardJS('.btn-copy', {
        target: function(trigger) {
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        let $btn = $(e.trigger);
        $btn.addClass('copied');
        let $icon = $($btn.find('i'));
        $icon.removeClass('ri-file-copy-2-line');
        $icon.addClass('ri-checkbox-circle-line');
        let $span = $($btn.find('span'));
        $span[0].innerText = 'COPIED';
        
        wait(function () { // 等待两秒钟后恢复
          $icon.removeClass('ri-checkbox-circle-line');
          $icon.addClass('ri-file-copy-2-line');
          $span[0].innerText = 'COPY';
        }, 2000);
      });
      clipboard.on('error', function(e) {
        e.clearSelection();
        let $btn = $(e.trigger);
        $btn.addClass('copy-failed');
        let $icon = $($btn.find('i'));
        $icon.removeClass('ri-file-copy-2-line');
        $icon.addClass('ri-time-line');
        let $span = $($btn.find('span'));
        $span[0].innerText = 'COPY FAILED';
        
        wait(function () { // 等待两秒钟后恢复
          $icon.removeClass('ri-time-line');
          $icon.addClass('ri-file-copy-2-line');
          $span[0].innerText = 'COPY';
        }, 2000);
      });
    }
    initCopyCode();
  }(window, document);
</script>


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